Etymology
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arrogate (v.)
"claim or demand presumptuously," 1530s, from Latin arrogatus, past participle of arrogare "to claim for oneself," from assimilated form of ad "to" (see ad-) + rogare "to ask, to propose (a law, a candidate); to ask a favor, entreat, request," apparently a figurative use of a PIE verb meaning literally "to stretch out (the hand)," from *rog-, variant of the root *reg- "move in a straight line." Related: Arrogated; arrogating.
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interrogative (adj.)
"asking or denoting a question," c. 1500, from Late Latin interrogativus "pertaining to a question," from interrogat-, past participle stem of Latin interrogare "to ask, question, inquire; interrogate judicially, cross-examine," from inter "between" (see inter-) + rogare "to ask, to question," apparently a figurative use of a PIE verb meaning literally "to stretch out (the hand)," from root *reg- "move in a straight line." As a noun, "word implying a question," 1520s. Related: Interrogatively.
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interrogation (n.)
late 14c., "a question;" c. 1500, "a questioning; a set of questions," from Old French interrogacion "a questioning" (13c.) or directly from Latin interrogationem (nominative interrogatio) "a question; questioning; judicial inquiry," noun of action from past participle stem of interrogare "to ask, question, inquire; interrogate judicially, cross-examine," from inter "between" (see inter-) + rogare "to ask, to question," apparently a figurative use of a PIE verb meaning literally "to stretch out (the hand)," from root *reg- "move in a straight line."
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mendicant (n.)

"a beggar, one who lives by asking alms," late 14c., from Latin mendicantem (nominative mendicans), noun use of present participle of mendicare "to beg, ask alms" (see mendicant (adj.)).

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askance (adv.)

1520s, "sideways, asquint, out of the corner of the eye," of obscure origin. OED has separate listings for askance and obsolete Middle English askance(s) and no indication of a connection, but Barnhart and others derive the newer word from the older one. The Middle English word, recorded early 14c. as ase quances and found later in Chaucer, meant "in such a way that; even as; as if;" and as an adverb "insincerely, deceptively." It has been analyzed as a compound of as and Old French quanses (pronounced "kanses") "how if," from Latin quam "how" + si "if."

The E[nglish] as is, accordingly, redundant, and merely added by way of partial explanation. The M.E. askances means "as if" in other passages, but here means, "as if it were," i.e. "possibly," "perhaps"; as said above. Sometimes the final s is dropped .... [Walter W. Skeat, glossary to Chaucer's "Man of Law's Tale," 1894]

Also see discussion in Leo Spitzer, "Anglo-French Etymologies," Philological Quarterly 24.23 (1945), and see OED entry for askance (adv.) for discussion of the mysterious ask- word cluster in English. Other guesses about the origin of askance include Old French a escone, from past participle of a word for "hidden;" Italian a scancio "obliquely, slantingly;" or that it is a cognate of askew.

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prerogative (n.)
Origin and meaning of prerogative

"special right or privilege granted to someone; characteristic right inhering in one's nature, office, or position," late 14c., prerogatif, (in Anglo-Latin from late 13c.), from Old French prerogative (14c.) and directly from Medieval Latin prerogativa "special right," from Latin praerogativa "prerogative, previous choice or election, privilege."

This was originally (with tribus, centuria) "unit of 100 voters who by lot voted first in the Roman comita," noun use of fem. of praerogativus (adj.) "chosen to vote first, that is asked before," from praerogere "ask before others," from prae "before" (see pre-) + rogare "to ask, ask a favor," apparently a figurative use of a PIE verb meaning literally "to stretch out (the hand)," from root *reg- "move in a straight line." In Middle English also "an innate faculty or property which especially distinguishes someone or something."

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interrogate (v.)
late 15c., a back-formation from interrogation or else from Latin interrogatus, past participle of interrogare "to ask, question, inquire; interrogate judicially, cross-examine," from inter "between" (see inter-) + rogare "to ask, to question," apparently a figurative use of a PIE verb meaning literally "to stretch out (the hand)," from root *reg- "move in a straight line." The Old French word was interroger (14c.) which yielded English interroge (late 15c.), now obsolete. Related: Interrogated; interrogating.
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requisition (n.)

c. 1400, requisicioun, "a request, an act of requesting or demanding," from Old French requisicion (12c.) and directly from Medieval Latin requisitionem (nominative requisitio) "examination, a searching," noun of action from past-participle stem of Latin requirere (past participle requisitus) "seek to know, ask, ask for" (see require).

The meaning "action of formally calling upon someone to perform some action, etc." is by 1550s, originally legal. The sense of "action of requiring a certain amount of something to be furnished" is by 1776.

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bid (v.)

probably an early Middle English mutual influence or confusion of two old words: The sense in bid farewell is from Old English biddan "to ask, entreat, beg, pray, beseech; order" (class V strong verb, past tense bæd, past participle beden), from Proto-Germanic *bedjanan "to pray, entreat" (source also of German bitten "to ask," attested in Old High German from 8c., also Old Saxon biddian, Old Frisian bidda"ask, request command," Old Norse biðja, Gothic bidjan "request"). This, according to Kluge and Watkins, is from a PIE root *gwhedh- "to ask, pray" (see bead (n.)).

To bid at an auction, meanwhile, is from Old English beodan "offer, proclaim" (class II strong verb; past tense bead, past participle boden), from Proto-Germanic *beudanan "to stretch out, reach out, offer, present," (source also of German bieten "to offer," Old High German biatan, also Old Saxon biodan, Old Frisian biada, Old Norse bjoða, Gothic anabiudan "to command"). This is (with a shift of meaning) from PIE root *bheudh- "be aware, make aware" (source also of bode (v.)).

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arrogance (n.)
"a manifest feeling of superiority of one's worth or importance, combined with contempt of others," c. 1300, from Old French arrogance (12c.), from Latin arrogantia "presumption, pride, haughtiness," abstract noun from arrogantem (nominative arrogans) "assuming, overbearing, insolent," present participle of arrogare "to claim for oneself, assume," from ad "to" (see ad-) + rogare "to ask, to propose (a law, a candidate); to ask a favor, entreat, request," apparently a figurative use of a PIE verb meaning literally "to stretch out (the hand)," from root *reg- "move in a straight line."
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